Problem: Simplify and expand the following expression: $ \dfrac{5}{3r - 27}- \dfrac{3}{r - 4}- \dfrac{3}{r^2 - 13r + 36} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the first term: $ \dfrac{5}{3r - 27} = \dfrac{5}{3(r - 9)}$ We can factor the quadratic in the third term: $ \dfrac{3}{r^2 - 13r + 36} = \dfrac{3}{(r - 9)(r - 4)}$ Now we have: $ \dfrac{5}{3(r - 9)}- \dfrac{3}{r - 4}- \dfrac{3}{(r - 9)(r - 4)} $ The least common multiple of the denominators is: $ 3(r - 9)(r - 4)$ In order to get the first term over $3(r - 9)(r - 4)$ , multiply by $\dfrac{r - 4}{r - 4}$ $ \dfrac{5}{3(r - 9)} \times \dfrac{r - 4}{r - 4} = \dfrac{5(r - 4)}{3(r - 9)(r - 4)} $ In order to get the second term over $3(r - 9)(r - 4)$ , multiply by $\dfrac{3(r - 9)}{3(r - 9)}$ $ \dfrac{3}{r - 4} \times \dfrac{3(r - 9)}{3(r - 9)} = \dfrac{9(r - 9)}{3(r - 9)(r - 4)} $ In order to get the third term over $3(r - 9)(r - 4)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{3}{(r - 9)(r - 4)} \times \dfrac{3}{3} = \dfrac{9}{3(r - 9)(r - 4)} $ Now we have: $ \dfrac{5(r - 4)}{3(r - 9)(r - 4)} - \dfrac{9(r - 9)}{3(r - 9)(r - 4)} - \dfrac{9}{3(r - 9)(r - 4)} $ $ = \dfrac{ 5(r - 4) - 9(r - 9) - 9} {3(r - 9)(r - 4)} $ Expand: $ = \dfrac{5r - 20 - 9r + 81 - 9}{3r^2 - 39r + 108} $ $ = \dfrac{-4r + 52}{3r^2 - 39r + 108}$